WASHINGTON — Michael D. Cohen, the former personal lawyer and fixer for President Trump, indefinitely postponed his congressional testimony set for next month, his lawyer said on Wednesday, depriving House Democrats of one of their first big spectacles in their plans to aggressively investigate the Trump administration.

Mr. Cohen’s lawyer Lanny J. Davis cited verbal attacks from Mr. Trump, who had begun suggesting after Mr. Cohen agreed to testify that one of his relatives be investigated for unspecified crimes.

Democrats called the president’s attacks “textbook mob tactics” of coercion. “The most upsetting thing about all of this is the fact that Mr. Cohen was intimidated,” Representative Elijah E. Cummings, Democrat of Maryland and chairman of the House Oversight Committee who had invited Mr. Cohen to testify, said in an interview. “And not only was he intimidated, but he believes his family also has been intimidated and threatened.”

Mr. Trump denied the charge, telling reporters at the White House that Mr. Cohen, who was sentenced last month to three years in prison for lying to Congress and other crimes, was “threatened by the truth.”

Mr. Cohen’s decision to back out of the hearing was a final retreat from what had initially promised to be a blockbuster hearing. He has said he does not want to be the “villain” of the Trump presidency and had originally pledged to give “a full and credible account” of his work for Mr. Trump to the House Oversight and Reform Committee, though lawmakers had warned that the special counsel, Robert S. Mueller III, would most likely keep him from discussing matters related to Russia.

Mr. Cohen’s deferral also put House Democrats in an awkward position. They must decide whether to force Mr. Cohen to testify despite his concerns about security or walk away altogether from a hearing that they held out as the first marquee moment in their plan to showcase their aggressive oversight of Mr. Trump and his administration.

Mr. Cummings said he would decide soon whether to subpoena Mr. Cohen to compel his testimony or if he will allow him to testify behind closed doors. Mr. Cohen is due to report to prison on March 6.

Privately, Mr. Trump’s aides had predicted that Mr. Cohen would back out of his appearance on Capitol Hill as Republicans prepared to raise questions about his character, including asking questions about his work before he joined the Trump Organization and his own antagonism toward reporters while he worked for Mr. Trump.

The president has repeatedly suggested on Twitter that Mr. Cohen’s family members be scrutinized. In a recent interview with Jeanine Pirro, the Fox News host and one of Mr. Trump’s preferred interviewers, he called for Mr. Cohen’s father-in-law to be investigated without citing details.

Mr. Trump has referenced the Eastern European background of Mr. Cohen’s father-in-law, Fima Shusterman, to imply that Mr. Shusterman has ties to organized crime. No evidence has emerged to back up that suggestion. Mr. Shusterman was sentenced to probation after pleading guilty in 1993, a year before Mr. Cohen married his daughter, to cashing $5.5 million in checks to evade federal reporting requirements. It is not clear that Mr. Trump is aware of that case.

A retiree, Mr. Shusterman emigrated to the United States in the early 1970s after facing anti-Jewish persecution in Soviet-controlled Ukraine, Mr. Cohen has told friends.

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Michael Cohen, President Trump’s former lawyer, pleaded guilty to lying to Congress about plans to build a tower for Mr. Trump in Russia. The admission is the latest twist in the onetime Trump loyalist’s deteriorating relationship with the president.CreditCreditChang W. Lee/The New York Times

Mr. Trump has also suggested the men have significant financial ties, which Mr. Cohen has denied. The president’s interview with Ms. Pirro prompted a rare statement from House Democrats cautioning that any effort to discourage or influence witness testimony before Congress could be construed as a crime.

The president’s allies in Congress dismissed the notion that his attacks constituted a threat to Mr. Cohen. “Everyone knows it wasn’t witness intimidation — c’mon,” Representative Jim Jordan of Ohio, the top Republican on the Oversight Committee, said in a brief interview. “If he comes, we’ll be ready. If he doesn’t, he doesn’t.”

Mr. Jordan and Representative Mark Meadows, Republican of North Carolina and another close ally of the president, indicated in a letter this week seeking documents from Mr. Cohen’s lawyer, Guy Petrillo, that his client would face aggressive questioning from House Republicans.

Mr. Trump’s allies have acknowledged privately for months that Mr. Cohen’s willingness to reveal the details of his work for Mr. Trump to prosecutors and the public stands as one of the most significant threats to the Trump presidency. In pleading guilty last year to helping arrange hush payments during the campaign to women who claimed to have had affairs with Mr. Trump, Mr. Cohen implicated the president, and prosecutors later said Mr. Trump directed the payments. The president has acknowledged that the money for the payments came from him after initially denying that he knew about them.

Mr. Cohen was also central to the Trump Organization proposal to build a skyscraper in Moscow, trying to secure permits for the plan during the 2016 presidential race. Mr. Mueller is scrutinizing the plan as part of his investigation into possible ties between Trump associates and Russia’s election interference.

Late last year, Mr. Cohen pleaded guilty to lying to Congress by testifying to congressional investigators that work on the project ended by January 2016. Prosecutors have said that negotiations continued until at least June of that year, and Mr. Trump’s lawyer Rudolph W. Giuliani acknowledged this weekend that they went on through the election, though he later sought to distance himself from that assertion.

The project has gained renewed attention in recent days — and Mr. Trump and his lawyer began attacking Mr. Cohen again — after a disputed report last week by BuzzFeed News said that Mr. Mueller had evidence that Mr. Trump personally directed Mr. Cohen to lie to Congress about the project. In a rare statement, a spokesman for Mr. Mueller called the article “inaccurate.”

Mr. Cohen has spent more than 70 hours in interviews with investigators for the Southern District of New York who prosecuted the campaign finance violations and for Mr. Mueller’s inquiry into Russia’s 2016 election interference and possible ties to the Trump campaign.

Once one of Mr. Trump’s loyal aides, Mr. Cohen fell out of favor with the president after pleading guilty in August to campaign finance violations and financial crimes. The president, who had urged Mr. Cohen to stay “strong” when federal agents raided his apartment, hotel room and office in April, switched gears and branded Mr. Cohen a convicted liar, weak and even a “rat.”

Mr. Davis compared Mr. Trump’s attacks on his client to a scene from the second “Godfather” movie and said that Mr. Trump was aware that Mr. Cohen’s “vulnerability” is his concern for his family.

The statements by Mr. Trump and Mr. Giuliani, who have accused Mr. Cohen of lying to get a reduced sentence, would most likely amount to sufficient evidence of witness tampering, according to legal experts, though they cautioned that prosecutors would bring such a case only if they viewed the tampering as part of a larger pattern of obstructive behavior. And a longstanding Justice Department view has held that a sitting president cannot be indicted.

For prosecutors to gain a conviction, they need to show only an attempt to influence testimony, not that the witness was actually deterred from speaking, said Chuck Rosenberg, a former senior Justice Department and F.B.I. official.

“Even if Cohen said the comments did not intimidate him, and that he would testify, those comments could still constitute witness tampering,” Mr. Rosenberg said. “The statute criminalizes the conduct of those who attempt to intimidate witnesses, not just those who succeed.”

A version of this article appears in print on , on Page A1 of the New York edition with the headline: Cohen Retreats From Testifying, Muffling Thunder of Democrats. Order Reprints | Today’s Paper | Subscribe